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You are given a deck of standard playing cards with 13 of [#permalink]
16 Jul 2003, 01:01

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A

B

C

D

E

Difficulty:

(N/A)

Question Stats:

0%(00:00) correct
0%(00:00) wrong based on 0 sessions

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You are given a deck of standard playing cards with 13 of the 52 cards designated as a тАЬheart.тАЭ You plan to shuffle the deck thoroughly then deal 10 cards off the top of the deck. What is the probability that the 10th card dealt is a heart?
(A) 1/4
(B) 1/5
(C) 5/26
(D) 12/42
(E) 13/42 _________________

1 heart in first nine (probably of 10th card being a heart is 12/43) or
2 hearts in first nine (probably of 10th card being a heart is 11/43) or
3 hearts in first nine (probably of 10th card being a heart is 10/43) or
4 hearts ....or 9 hearts in first nine.

I can't seem to figure out the answer though. Because I get 85 in the numerator....

The correct answer is (A). Although this may be counter-intuitive at first, the probability of any card in the deck being a heart before any cards are seen is 1/4. We can also solve this analytically for any card by building a probability тАЬtreeтАЭ and summing the probability of all of its тАЬbranches.тАЭ

For example, letтАЩs find the probability that the 2nd card is a heart. There are two mutually exclusive ways that can happen: (1) both the first and second cards are hearts; and (2) only the second card is a heart.
Using the multiplication rule, the probability that the first card is a heart AND the second card is a heart is equal to the probability of picking a heart on the first card or 13/52 (number of hearts in a full deck divided by number of cards in the deck) times the probability of subsequently picking a heart on the 2nd card or 12/51 (number of hearts remaining in the deck divided by number of cards remaining in the deck) which equals 12/204.

Similarly, the probability that the first card is a non-heart AND that the second card is a heart is equal to the probability that the first card in NOT a heart or 39/52 (number of non-hearts in a new deck divided by number of cards in the deck) times the probability of subsequently picking a heart on the 2nd card or 13/51 (number of hearts in the deck divided by the number of cards remaining in the deck) or 39/204.

Since these two events are mutually exclusive, we can add them together to get the total probability of getting a heart as the second card: i.e., 12/204 + 39/204 = 51/204 = 1/4. We can do a similar analysis for any card in the deck, and, although the probability tree gets more complicated as the card number gets higher, the total probability of the nth card being a heart will always end up simplifying to 1/4. _________________